Collective Behavior of Polar Impurities in Ionic Crystals. II

The purpose of this work is to construct an improved theory to account for the low-temperature dielectric properties of KCl crystals containing substitutional ${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}^{-}$ impurities. The statistical mechanics of a pair of dipoles is discussed in order to derive a generalized Langevin function giving the mean moment of such a pair along the direction of an applied field. A hybrid approximation is then developed, in which the interaction of a typical pair of ${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}^{-}$ dipoles is treated explicitly while the remaining interactions are treated by means of a modified Onsager-Pirenne approximation. The resulting equations predict a second-order phase transition at $\frac{k{T}_c}{N{m}^2}\cong 0.53$, where $N$ is the concentration and $m$ the dipole moment of the ${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}^{-}$ ions. The predicted spontaneous polarization increases continuously below $T_c$ to reach a saturation value of ${P_0}^{max}\cong 0.45Nm$. A thermodynamic analysis indicates that the asymptotic behavior of the model as $T\to 0$ is essentially in agreement with the third law, and also enables one to show that the critical temperature should be independent of the applied field.